3.39 \(\int \frac{1}{x \sqrt [3]{2-3 x+x^2}} \, dx\)

Optimal. Leaf size=110 \[ \frac{3 \log \left (-2^{2/3} \sqrt [3]{x^2-3 x+2}-x+2\right )}{4 \sqrt [3]{2}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{2} (2-x)}{\sqrt{3} \sqrt [3]{x^2-3 x+2}}+\frac{1}{\sqrt{3}}\right )}{2 \sqrt [3]{2}}-\frac{\log (2-x)}{4 \sqrt [3]{2}}-\frac{\log (x)}{2 \sqrt [3]{2}} \]

[Out]

-(Sqrt[3]*ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - x))/(Sqrt[3]*(2 - 3*x + x^2)^(1/3))])/(2*2^(1/3)) - Log[2 - x]/(4*2
^(1/3)) - Log[x]/(2*2^(1/3)) + (3*Log[2 - x - 2^(2/3)*(2 - 3*x + x^2)^(1/3)])/(4*2^(1/3))

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Rubi [A]  time = 0.0249725, antiderivative size = 176, normalized size of antiderivative = 1.6, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {755, 123} \[ \frac{3 \sqrt [3]{x-2} \sqrt [3]{x-1} \log \left (-\frac{(x-2)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{x-1}\right )}{4 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac{\sqrt [3]{x-2} \sqrt [3]{x-1} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}}-\frac{\sqrt{3} \sqrt [3]{x-2} \sqrt [3]{x-1} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\sqrt [3]{2} (x-2)^{2/3}}{\sqrt{3} \sqrt [3]{x-1}}\right )}{2 \sqrt [3]{2} \sqrt [3]{x^2-3 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(2 - 3*x + x^2)^(1/3)),x]

[Out]

-(Sqrt[3]*(-2 + x)^(1/3)*(-1 + x)^(1/3)*ArcTan[1/Sqrt[3] - (2^(1/3)*(-2 + x)^(2/3))/(Sqrt[3]*(-1 + x)^(1/3))])
/(2*2^(1/3)*(2 - 3*x + x^2)^(1/3)) + (3*(-2 + x)^(1/3)*(-1 + x)^(1/3)*Log[-((-2 + x)^(2/3)/2^(1/3)) - 2^(1/3)*
(-1 + x)^(1/3)])/(4*2^(1/3)*(2 - 3*x + x^2)^(1/3)) - ((-2 + x)^(1/3)*(-1 + x)^(1/3)*Log[x])/(2*2^(1/3)*(2 - 3*
x + x^2)^(1/3))

Rule 755

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[((b + q + 2*c*x)^(1/3)*(b - q + 2*c*x)^(1/3))/(a + b*x + c*x^2)^(1/3), Int[1/((d + e*x)*(b + q + 2*c
*x)^(1/3)*(b - q + 2*c*x)^(1/3)), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c^2*d^2 -
b*c*d*e - 2*b^2*e^2 + 9*a*c*e^2, 0]

Rule 123

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[
(b*(b*e - a*f))/(b*c - a*d)^2, 3]}, -Simp[Log[a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[(Sqrt[3]*ArcTan[1/Sqrt[3
] + (2*q*(c + d*x)^(2/3))/(Sqrt[3]*(e + f*x)^(1/3))])/(2*q*(b*c - a*d)), x] + Simp[(3*Log[q*(c + d*x)^(2/3) -
(e + f*x)^(1/3)])/(4*q*(b*c - a*d)), x])] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt [3]{2-3 x+x^2}} \, dx &=\frac{\left (\sqrt [3]{-4+2 x} \sqrt [3]{-2+2 x}\right ) \int \frac{1}{x \sqrt [3]{-4+2 x} \sqrt [3]{-2+2 x}} \, dx}{\sqrt [3]{2-3 x+x^2}}\\ &=-\frac{\sqrt{3} \sqrt [3]{-2+x} \sqrt [3]{-1+x} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{\sqrt [3]{2} (-2+x)^{2/3}}{\sqrt{3} \sqrt [3]{-1+x}}\right )}{2 \sqrt [3]{2} \sqrt [3]{2-3 x+x^2}}+\frac{3 \sqrt [3]{-2+x} \sqrt [3]{-1+x} \log \left (-\frac{(-2+x)^{2/3}}{\sqrt [3]{2}}-\sqrt [3]{2} \sqrt [3]{-1+x}\right )}{4 \sqrt [3]{2} \sqrt [3]{2-3 x+x^2}}-\frac{\sqrt [3]{-2+x} \sqrt [3]{-1+x} \log (x)}{2 \sqrt [3]{2} \sqrt [3]{2-3 x+x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0252075, size = 59, normalized size = 0.54 \[ -\frac{3 \sqrt [3]{1-\frac{2}{x}} \sqrt [3]{1-\frac{1}{x}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{1}{x},\frac{2}{x}\right )}{2 \sqrt [3]{x^2-3 x+2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(x*(2 - 3*x + x^2)^(1/3)),x]

[Out]

(-3*(1 - 2/x)^(1/3)*(1 - x^(-1))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, x^(-1), 2/x])/(2*(2 - 3*x + x^2)^(1/3))

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt [3]{{x}^{2}-3\,x+2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^2-3*x+2)^(1/3),x)

[Out]

int(1/x/(x^2-3*x+2)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - 3 \, x + 2\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-3*x+2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((x^2 - 3*x + 2)^(1/3)*x), x)

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Fricas [B]  time = 18.5736, size = 828, normalized size = 7.53 \begin{align*} -\frac{1}{12} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3} 2^{\frac{1}{6}}{\left (2^{\frac{5}{6}}{\left (x^{6} + 36 \, x^{5} - 612 \, x^{4} + 2880 \, x^{3} - 5760 \, x^{2} + 5184 \, x - 1728\right )} + 12 \, \sqrt{2}{\left (x^{5} - 38 \, x^{4} + 252 \, x^{3} - 648 \, x^{2} + 720 \, x - 288\right )}{\left (x^{2} - 3 \, x + 2\right )}^{\frac{1}{3}} + 48 \cdot 2^{\frac{1}{6}}{\left (x^{4} - 6 \, x^{3} + 6 \, x^{2}\right )}{\left (x^{2} - 3 \, x + 2\right )}^{\frac{2}{3}}\right )}}{6 \,{\left (x^{6} - 108 \, x^{5} + 972 \, x^{4} - 3456 \, x^{3} + 6048 \, x^{2} - 5184 \, x + 1728\right )}}\right ) + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (\frac{2^{\frac{2}{3}} x^{2} + 6 \cdot 2^{\frac{1}{3}}{\left (x^{2} - 3 \, x + 2\right )}^{\frac{1}{3}}{\left (x - 2\right )} + 12 \,{\left (x^{2} - 3 \, x + 2\right )}^{\frac{2}{3}}}{x^{2}}\right ) - \frac{1}{24} \cdot 2^{\frac{2}{3}} \log \left (\frac{12 \cdot 2^{\frac{2}{3}}{\left (x^{2} - 3 \, x + 2\right )}^{\frac{2}{3}}{\left (x^{2} - 6 \, x + 6\right )} + 2^{\frac{1}{3}}{\left (x^{4} - 36 \, x^{3} + 180 \, x^{2} - 288 \, x + 144\right )} - 6 \,{\left (x^{3} - 14 \, x^{2} + 36 \, x - 24\right )}{\left (x^{2} - 3 \, x + 2\right )}^{\frac{1}{3}}}{x^{4}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-3*x+2)^(1/3),x, algorithm="fricas")

[Out]

-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^6 + 36*x^5 - 612*x^4 + 2880*x^3 - 5760*x^2 + 5184
*x - 1728) + 12*sqrt(2)*(x^5 - 38*x^4 + 252*x^3 - 648*x^2 + 720*x - 288)*(x^2 - 3*x + 2)^(1/3) + 48*2^(1/6)*(x
^4 - 6*x^3 + 6*x^2)*(x^2 - 3*x + 2)^(2/3))/(x^6 - 108*x^5 + 972*x^4 - 3456*x^3 + 6048*x^2 - 5184*x + 1728)) +
1/12*2^(2/3)*log((2^(2/3)*x^2 + 6*2^(1/3)*(x^2 - 3*x + 2)^(1/3)*(x - 2) + 12*(x^2 - 3*x + 2)^(2/3))/x^2) - 1/2
4*2^(2/3)*log((12*2^(2/3)*(x^2 - 3*x + 2)^(2/3)*(x^2 - 6*x + 6) + 2^(1/3)*(x^4 - 36*x^3 + 180*x^2 - 288*x + 14
4) - 6*(x^3 - 14*x^2 + 36*x - 24)*(x^2 - 3*x + 2)^(1/3))/x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt [3]{\left (x - 2\right ) \left (x - 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**2-3*x+2)**(1/3),x)

[Out]

Integral(1/(x*((x - 2)*(x - 1))**(1/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{2} - 3 \, x + 2\right )}^{\frac{1}{3}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-3*x+2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((x^2 - 3*x + 2)^(1/3)*x), x)